A006772 Sum of spans of 2n-step polygons on square lattice.

0, 1, 3, 14, 70, 370, 2028, 11452, 66172, 389416, 2326202, 14070268, 86010680, 530576780, 3298906810, 20653559846, 130099026600, 823979294284, 5244162058026, 33523117491920, 215150177410088, 1385839069134800, 8956173544332434, 58056703069399056, 377396656568011618, 2459614847765495754, 16068572108927106202

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			From _Andrey Zabolotskiy_, Nov 09 2018: (Start)
There are no 2-step polygons (conventionally).
For n=2, the only 4-step polygon is a 1 X 1 square having span 1, so a(2)=1.
For n=3, the only 6-step polygon is a 2 X 1 domino which can be rotated 2 ways having spans 2 and 1, so a(3) = 2+1 = 3.
For n=4, there are the following 8-step polygons:
a 3 X 1 stick which can be rotated 2 ways having spans 3 and 1;
an L-tromino which can be rotated 4 ways, all having span 2;
a 2 X 2 square, having span 2.
So a(4) = 3 + 1 + 4*2 + 2 = 14.
For n=5, there are the following 10-step polygons:
a 4 X 1 stick which can be rotated 2 ways having spans 4 and 1;
an L-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3, plus reflections;
a T-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3;
an S-tetromino which can be rotated 2 ways having spans 3 and 2, plus reflections;
a 3 X 2 rectangle which can be rotated 2 ways having spans 3 and 2;
a 3 X 2 rectangle without one of its angular squares having same counts as L-tetromino.
So a(5) = 4 + 1 + 2 * 2*2*(2+3) + 2*(2+3) + 2*(3+2) + 3 + 2 = 70.
(End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.

Crossrefs

Cf. A002931, A006773, A302336, A302337, A232103.

Extensions

Name corrected, more terms from Andrey Zabolotskiy, Nov 09 2018